180 XIII INTEGRATION
(b) The function g is measurable, and for each compact set K c X we
have
I
*
\d\9vi dfj,< +00.
(c) For each function h e Jfc(X), the function gh is integrable.
Since we may write g — g± 4- ig2 > where gl and g2 are K- valued, and then
9i =0i -9l> 92=92 ~92, we reduce straightaway ((13.9.6) and (13.10))
to the case where g is a mapping ^0 of X into R. To show that (a) implies
(b), cover K by a finite number of open sets V,- such that g<pVj is integrable for
each y. Then sup(#<pv.,) is integrable (13.7.4) and hence so is
(13.9.14). Hence (b) follows from (13.9.13). To show that (b) implies (c),
observe first that g is almost everywhere finite (X being a denumerable
union of compact sets), hence gh is measurable (13.9.8.1). Moreover, if
L = Supp(/z), we have \gh\ ^ \g<pi\ • \\h\\, hence it follows from (b) and
(13.9.13) that gh is integrable. Finally, to show that (c) implies (a), consider a
compact neighborhood V of x e X and a continuous mapping h : X -» [0, 1]
which is equal to 1 on V and has compact support ((3.18.2) and (4.5.2)). By
hypothesis gh is integrable, hence so is gcpv = (gh)(pv (13.9.14).
When the equivalent conditions of (13.13.1) are satisfied, the function g
is said to be locally integrable (with respect to ju) or locally fji-integrable.
Clearly every integrable function is locally integrable. Every measurable
function whose restriction to every compact subset of X is bounded almost
everywhere (in particular, every function belonging to & £ or JSf £) is locally
integrable. Every function belonging to :$?£ or jSf £ is locally integrable, by
(13.11.7). Every function equivalent to a locally integrable function is locally
integrable. We have remarked in the course of the proof of (13.13.1) that
every locally integrable function is finite almost everywhere. The function
on R which is equal to l/\x\ when x ^ 0 and is 0 when x = 0 is lower semi-
continuous but not locally integrable with respect to Lebesgue measure.
For a complex- valued function g to be locally integrable, it is necessary and
sufficient that 0tg and Jg should be locally integrable. For a real-valued
function g to be locally integrable, it is necessary and sufficient that g+
and g~~ should be locally integrable.
Let g be a locally ^-integrable function. Since /H* \fg d^ is defined on the
whole of Jfc(X), it is a linear form on this complex vector space. Moreoverne or other of the partitions V u~J(oc).